Bounded, attractive and repulsive Markov specifications on trees and on the one-dimensional lattice

Stan Zachary

Research output: Contribution to journalArticle

Abstract

Let ? be a homogenous Markov specification associated with a countable state space S and countably infinite parameter space A possessing a neighbor relation ~ such that (A,~) is the regular tree with d +1 edges meeting at each vertex. Let g(p)be the simplex of corresponding Markov random fields. We show that if ? satisfies a 'boundedness' condition then g(p).We further study the structure of g(p) when ? is either attractive or repulsive with respect to a linear ordering on S. When d = 1, so that (A, ~) is the one-dimensional lattice, we relax the requirement of homogeneity to that of stationarity; here we give sufficient conditions for g(p) and for g(p)to have precisely one member. © 1985.

Original languageEnglish
Pages (from-to)247-256
Number of pages10
JournalStochastic Processes and their Applications
Volume20
Issue number2
Publication statusPublished - Sep 1985

Keywords

  • attractive specifications
  • Markov chains on infinite trees
  • Markov random fields
  • phase transition
  • repulsive specifications

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